A Bernstein-type theorem for $\xi$-submanifolds with flat normal bundle in the Euclidean spaces

Authors: Xu-Yong Jiang, HE-JUN SUN, PEIBIAO ZHAO

Abstract: $\xi$-Submanifolds in the Euclidean spaces are a natural extension of self-shrinkers and a generalization of $\lambda$-hypersurfaces. Moreover, $\xi$-submanifolds are expected to take the place of submanifolds with parallel mean curvature vector. In this paper, we establish a Bernstein-type theorem for $\xi$-submanifolds in the Euclidean spaces. More precisely, we prove that an $n$-dimensional smooth graphic $\xi$-submanifold with flat normal bundle in $\mathbb{R}^{n+p}$ is an affine $n$-plane.

Keywords: $\xi$-Submanifold, Bernstein-type theorem

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